I am reading up a proof of Hensel's lemma over the $p$-adic integers and found a proof here, starting on page $8$ which I feel is the most accessible...well sort of anyway.
My groundings on analysis are quite weak but I think I understand the reason as to why the author wants to construct a Cauchy sequence as they always converge right? I think that's all fine and well - but I get a bit lost as to why the sequence that is constructed in the proof is automatically Cauchy.
Any clarification on this matter will be greatly appreciated, because I think the rest of the proof is fine.
Thanks in advance.
For the $p$-adic norm, to show that a sequence $\{a_n\}$ is Cauchy, so for all $\epsilon>0$, there exists an $n_0$ such that, for all $m>n\ge n_0$, one has $$|a_m-a_n|_p < \epsilon,$$ it is sufficient to show that there exists an $n_0$ such that, for all $n\ge n_0$, one has $$|a_{n+1}-a_n|_p < \epsilon,$$ since $$|a_m-a_n|_p \le \max_{m> i\ge n}\{|a_{i+1}-a_i|_p\}.$$ Euqivalently, one has to show that $$\lim_{n\to \infty} |a_{n+1}-a_n|_p=0.$$
But, for the way they are constructed, $a_{n+1}\equiv a_{n}\pmod{p^n}$, so $$|a_{n+1}-a_n|_p=|t_np^n|_p=|t_n|_p/p^n$$ where $t_n\in \mathbb{Z}_p$, so $|t_n|_p\le 1$. Hence $$\lim_{n\to \infty} |a_{n+1}-a_n|_p=\lim_{n\to \infty}|t_n|_p/p^n=0.$$